23,506 research outputs found
Recovering Cusp forms on GL(2) from Symmetric Cubes
Suppose , are cusp forms on GL, not of solvable polyhedral
type, such that they have the same symmetric cubes. Then we show that either
, are twist equivalent, or else a certain degree -function
associated to the pair has a pole at . If we further assume that the
symmetric fifth power of is automorphic, then in the latter case,
is icosahedral in a suitable sense, agreeing with the usual notion when there
is an associated Galois representation.Comment: 10 page
Messaging and Public Opinion on Immigration Reform
We started off the survey by asking respondents to rank the seriousness of the issue of illegal immigration. Respondents were first asked to rate the seriousness of the issue with respect to the United States, and then in relation to their city or community
Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2)
A fundamental question, first raised by Langlands, is to know whether the
Rankin-Selberg product of two (not necessarily holomorphic) cusp forms f and g
is modular, i.e., if there exists an automorphic form f box g on GL(4)/Q whose
standard L-function equals L^*(s, f x g) after removing the ramified and
archimedean factors. The first main result of this paper is to answer it in the
affirmative, in fact with the base field Q replaced by any number field F. Our
proof uses a mixture of converse theorems, base change and descent, and it also
appeals to the local regularity properties of Eisenstein series and the scalar
products of their truncations.
One of the applications of this result is that the space of cusp forms on
SL(2) has multiplicity one. Concretely this means the following: If f, g are
newforms of holomorphic or Maass type with trivial character such that the
squares of the p-th coeficients of f and g are the same at almost all primes p,
then g must be a twist of f by a quadratic Dirichlet character.Comment: 67 pages, published version, abstract added in migratio
Icosahedral Fibres of the Symmetric Cube and Algebraicity
For any number field F, call a cusp form π = π_∞⊗πf on GL(2)/F special icosahedral, or just s-icosahedral for short, if π is not solvable polyhedral, and for a
suitable “conjugate” cusp form π' on GL(2)/F, sym^3(π) is isomorphic to sym^3(π'), and the symmetric fifth power L-series of π equals the Rankin-Selberg L-function
L(s, sym^2(π') × π) (up to a finite number of Euler factors). Then the point of this Note is to obtain the following result: Let π be s-icosahedral (of trivial central character). Then π f is algebraic without local components of Steinberg type, π ∞ is of Galois type, and π_v is tempered every-where. Moreover, if π' is also of trivial central character, it is s-icosahedral, and the
field of rationality Q(πf) (of πf) is K := Q[√5], with π'
_f being the Galois conjugate of πf under the non-trivial automorphism of K. There is an analogue in the case of non-trivial central character ω, with the conclusion that π is algebraic when ω is, and when ω has finite order, Q(πf) is contained in a cyclotomic field
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